Optimal regulation of hyperbolic systems in the presence of unknown disturbances is considered. Necessary conditions for determining the optimal control that tracks a desired trajectory in the presence of the worst possible perturbations are developed. The results also characterize the worst possible disturbance that the system will be able to tolerate before any degradation of the system performance. Numerical results on the control of a vibrating beam are presented

Ahmed, N. U.

Biswas, Saroj K.

English

NASA Technical Reports Server

N94- 35882
OPTIMAL DISTURBANCE REJECTING CONTROL
OF IIYPERBOLIC SYSTEMS
Saroj K. Biswas
Department of Electrical Engineering
Temple University
Philadelphia, PA
N. U. Ahmed
Department of Electrical Engineering
University of Ottawa
Ottawa, Ontario, Canada
f
ABSTRACT
Optimal regulation of hyperbolic systems in the presence of unknown disturbances is considered.
Necessary conditions for determining the optimal control that tracks a desired trajectory in the presence
of the worst possible perturbations are developed. The results also characterize the worst possible
disturbance that the system will be able to tolerate before any degradation of the system performance.
Numerical results on the control of a vibrating beam are presented.
I. INTRODUCTION
The Hoo control problem for regulation of dynamical systems in the presence of perturbations has
been a subject of considerable research in recent years [4,5,6,7,10,16]. Although the original formulation
of the H_ method was in terms of frequency domain terms, extensions in state space terms leading
to feedback control using Riccati type equations have been developed [4,7,10,16]. For the infinite
dimensional systems, the Hoo control has started to gain momentum. For a summary of recent results,
see the survey paper [3]. Like its finite dimensional counterpart, the frequency domain approach [9,15] as
well as the state space analysis [11,12,] in the presence of both bounded and unbounded perturbations
has been considered in the literature. The problems pertinent to the H_ control design are: a)
input-output stability, b) disturbance decoupling, and c) disturbance attenuation.
This paper is concerned with disturbance attenuation of hyperbolic systems in the presence of worst
possible disturbances. We utilize the concepts of optimal control theory [1,8] for infinite dimensional
systems for deriving the control law for optimum regulation of the system in the presence of worst
possible disturbances. The method presented in this paper is a generalization of an H_-type method
developed in [13,14] for finite dimensional systems. The ratio of disturbance energy to the energy of
the controlled system is used as a measure of performance for disturbance attenuation. We present
conditions for estimation of the largest perturbation that can be attenuated and the corresponding
controller to attenuate this perturbation.
The paper is organized as follows: Section II introduces the H_ control problem. Necessary
conditions for optimum disturbance attenuation are presented in section III followed by numerical
results on control of a vibrating beam in section IV. Some concluding remarks are given in section V.
II. NOTATIONS AND PROBLEM STATEMENT
We shall use the following notations for abstract function spaces throughout the paper. Let H be a
Hilbert space, and V a linear subspace of H carrying the structure of a reflexive Banach space with the
injection V_H continuous. We identify H with its dual so that V_HC_V', where V' is the topological
dual of V.
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https://ntrs.nasa.gov/search.jsp?R=19940031375 2020-06-16T11:11:30+00:00Z
SupposeA be a bounded linear self adjoint operator A E £(V,V') satisfying the conditions
I(A_,0)I < cll_llv IlOIIv,c > 0, _,_ _ v
<A_,_>v,,v + Zl_lzz >__11_112,,, > O,a e R, _ e V" (1)
Consider the hyperbolic system
02y
Or------£ + Ay = Bu + Cv, tEI=(0, T),
y(o) = yo, _(o) = v_
of
(2)
where the operator A is as defined above. The control applied to the system is denoted u E H - L2(I, H),
and B is a bounded linear operator B E £(H). Suppose the system is perturbed by a disturbance
v E L2([, H) through the operator C E £(H). The initial conditions y0 E V and y_) E H are also
considered to be initial disturbances to the system.
With this introduction we now pose the control problem:
Given the perturbed system (2), find the control u E L2(I,H) that keeps the state trajectory as
close as possible to a desired trajectory in the presence of maximum possible additive disturbance
v E L2(I, H) and maximum possible initial disturbances y0 E V and y6 E H.
For a mathematical formulation of this control problem, we introduce a cost function:
½sx ff_ ly012dx + ½s2 ff_ [y_l2 dx + ½ fI×f_ r21vl 2 dx dt
J(u,v, yo,yh) = ½q_fz×f_ly- YalZdxdt + ½q2fz×f_lY,-Y_I 2dzdt + ½fl×f_nlul 2dxdt (3)
where Sl,s2, ql,q2, rl and r2 are scalar weighting factors, and yd and y] are desired trajectories respec-
tively. Then the disturbance rejecting control problem is equivalent to the minimax problem of finding
a control u and a scalar A" so that
A* = inf sup J(u, v, Yo, Y_o) (4)
v_0 uELt
y0_0
y_0
subject to the dynamics (2). The quantity A" can be interpreted as the disturbance rejection capacity of
the system. A larger A" implies a better controller in the sense that the system will be able to tolerate
larger amount perturbations before degradation of the system performance. A small A" means that the
system is too sensitive to disturbances; despite the effects of the control the state trajectory is not close
to the desired trajectory even in the presence of a small amount of perturbations.
We shall assume that a solution of this minimax problem exists. In what follows, we shall derive a
set of necessary conditions that must be satisfied by the optimal controller.
III. MAIN RESULTS
We first give a brief outline of derivation of the main results. The minimax problem introduced
above is solved in two steps, with the first step being finding the supremum of J over u assuming
that the perturbations v and y0, y6 are known, and the second finding the infimum of J over nonzero
perturbations. The first step determines the optimal control that regulates the system, and the second
step characterizes the worst possible perturbation that the controller will be able to attenuate before a
serious degradation of the system performance.
318
Clearly,the problemof findingthe supremumof J over u is equivalent to minimizing the denomi-
nator of J given in (3) for flExed v, Y0, and y6 subject to the dynamics (2). This is a well known problem
in infinite dimensional control theory for hyperbolic systems (see [1,2,8] for details). Theorems 1 and 2
presented below pertain to this problem. We omit the proofs for brevity.
TIIEOREM 1. For a given v E Le(I,H), Yo E V, and Y_oE H, the system (1) has a unique solution
y E L2(I, V) M C(], V), yt E L2(I, H) n C(], H). Furthermore, the mapping (yo, Y'o, u) -_ (y, yt) is
continuous from V x H x L2(I, H) _ L2(I, V) x L2([, H). •
For convenience of presentation, we introduce two new variables _1 = y and _2 _ Yt, and rewrite
the system (2) as a first order equation:
04
0--7+ A_ = 13u + Cv (5)
= - p
where [0o'] [o] [o]A = A , 13 = and C = .
Using the above notations, we also rewrite the denominator of the cost function (3) as
J10t) = "_ (_9 - _d, Q(_ _ _d))H× H dt -b _ (u, Rl_t)t f dt (6)
where Q = diag(ql,q2) and R1 = rl. Let ql and q2 be nonnegative, and rl strictly positive.
The necessary and sufficient condition that uo E L2(I,H) be optimal in the sense of minimization
of the cost (6) is that
J_(u0;u-u0)>_0 for alluEH (7)
where J{(u0; u - u0) is the Gateaux derivative of J1 at u0 E H in the direction u - u0. This is given in
the next theorem:
TttEOREM 2. Consider the system (5)for ffvced additive disturbance v E L2(I, H), ffLred initial disturbance
p E V x 1t, and the desired trajectory Ca E L2(I, H) x L2(I, H). Then the optimal control uo E L2(I, H)
that minimizes the cost (6) is characterized by the sohaion of the two-point-boundary-vah_e problem."
O_
0-_- + A_ + _RIÂ_*_ = Cv, _(0) = p (8)
0_ +A*
- 0--t _P = Q(_ - _d), g,(T) = 0 (9)
The optimal control uo is then given by
uo = -R-{1B* _/,. • (10)
At this point we return to the disturbance rejecting control problem introduced earlier. Clearly J(u0)
is a function of v and p which are yet to be determined. We substitute Jl(u0) into the denominator of
the cost function (3) leading to
i(p,sp). + ½
= (11)
J(v,p) ! fI(_--?d,O(_--_d))H2 x Hdt + ½fi(BR-{1B*_",g') dt
319
Then the disturbance rejecting control problem is equivalent to finding a scalar A* so that
A* = inf J(v,p) (12)
v_0
pC:0
subject to the dynamics (8) - (9). Note that since A* is the optimal solution, we have 0 < A* < A = J(v, p).
Hence clearly, the following function
J2(v,p) = _(p, Sp)H+ _ (v, R2v)Hdt- -_ (_--_d,Q(c2--_d))HxHdt-- -_ (BR-{1B*_,z_,)dt (13)
is convex and nonnegative, and has a minimum at J2 = 0. Thus the problem of finding the infimum
indicated in (12) is equivalent to minimizing (13) subject to the system dynamics (8) - (9).
By virtue of Theorem 1, it is clear that for any v E L2(I, H), p E l," × tl, and _d E L2(I, H) × L2(I, H),
the equations (8) - (9) have a unique solution _ E L2(I, V) × L2(I, H) and _ E L2(I, V) × L2(I, H). In
addition, the solution has a unique Gateaux derivative satisfying the following theorem:
THEOREM 3. The solution (_,_)of the two-point-boundary-value problem (8) - (9)con'esponding to
v E L2(I,H) and p E V × H has a unique Gateaux derivative at every vo E L2(I,H) and po E V x H
satisfying
o_
0--7+ .a_ + _R?_'_, = C(v - vo),
_o_ + A*_,- Q_ = o, _,(T) = 0
Ot
with _ E L2(I, V) × L2(I, H) and _b E L2(I, V) × L2(I, H) •
_(o) = p - po (14)
(15)
Necessary conditions of optimality for minimization of (13)are now derived with the help of the
above results and the fact that the Gateaux derivative
J_(v0, P0; v - v0, p - P0) _> 0 (16)
for all v E L2(I,H) and p0 E V x H, where J_ is the Gateaux derivative at vo,po in the direction
v - v0, p - p0. We present the result in the following theorem:
THEOREM 4.
by the optimal control uo are characterized by simultaneous solution of the following equations:
The worst additive disturbance vo and the worst initial disturbance po that can be attenuated
-- = c/_2 c _, (17)Ot + .d_ + ]3RllB'_ , -1 ,.
o_ A"
- 0--7+ e, = Q(_ - _d), (18)
o(
- 0--[ + .A*( - Qr/= A'Q(v_ - v_d) (19)
O_
0--/ + Mr/ + BRllB*_ -- A*I3RllI3"g, (20)
_(0) = p0
_/,(T) = 0
_(o) = Spo
_(T) = 0
r/(o) = o
with the boundary conditions
(21)
320
The worst disturbances are given by
vo = R2aC*_ (22)
p0 = s-l (0)
The optimal control that regulates the system in the presence of the worst disturbance is given by
u0 = -Rll_*_ (23)
PRoof: Taking the Gateaux derivative of the Jz at v0 E L2(I, H) and p0 E V x H, we have
: /, J,0 <_ J2(_o,Po, v-vo,p-po) (Spo, p-po)H+ (R2vo, v-vo) dt-A* (_,O(q___a)) dt-A* (_,,t_R 1 1BO)dt
The result follows from Theorem (3) and adjoint system (19) - (20). •
It is worthwhile to mention here that equations (17) - (21) represent a two-point-boundary-value
problem with A* as a parameter which is unknown. The smallest value of A for which this TPBVP, i.e.,
(17) - (21) has a solution is the optimal A* or the disturbance rejection capacity of the system. The
corresponding control u0 is then obtained using (23) and the worst disturbance v0 and p0 that can be
attenuated is given by (22).
IV. EXAMPLE
We consider the cantilever beam equation (normalized)
02y 04y
Ot--'--_ + OX 4 -- g(X)U(t) + h(x)v(t), x E f_ = (0, 1), t > 0 (24)
subject to boundary conditions
_x (1,t_ (1 t) = 0 (25)
02y 03y
y(o,t) = o, (o, t) = o, , = o, ,
Define the operator A in H = Le(gt) by
{ 030.}04¢ D(A) = 4, : ¢ E H4(a),¢(0) = 0, (0) = 0, _-_z2(1) = 0, _-ff(U = 0A(_- OX4,
where H4(Q) is the Sobolev space of order four on _. For V we take V = {¢ E HZ(f_),¢(0) =
0,
We assume that the desired state of the controlled system is the zero state, and that there is no
initial disturbance. We compute the disturbance rejection capacity of the system using Theorem 4.
Table I shows that a tighter regulation (i.e., higher Q) is possible only if less disturbance is allowed
to be attenuated. It is intuitively correct to say that a better regulation of the state trajectory can
be achieved if the disturbance amplitude is small. Similarly a cheaper control allows more disturbance
accommodation by the controller as shown in Table II. Stated in a different way, this means that
attenuation of larger amplitude disturbances will require more control energy.
TABLE I TABLE II
Q rl r2 S A*
1 1 1 10 0.5522
10 1 1 10 0.2130
20 1 1 10 0.1521
50 1 1 10 0.0922
100 1 1 10 0.0618
Q rl r2 S A*
20 1 1 10 0.1521
20 2 1 10 0.1065
20 5 1 10 0.0637
20 10 1 10 0.0420
321
V. CONCLUSION
We present an H_o-like control method for hyperbolic systems. Necessary conditions in the form of
a two-point-boundary-value problem for determining the optimum controller and the worst exogenous
input that can be attenuated by the optimum controller have been derived. The results are related to
the H¢¢ control problem in the sense that the H_ norm is given by the inverse of square root of £*
[14]. The disturbance rejection capacity has been computed for a cantilever beam. Further research
needs to be done to develop state feedback and output feedback controllers, and to extend the method
to the infinite horizon control problems.
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Optimal disturbance rejecting control of hyperbolic systems

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